Trapezoid
"Midsegment and Diagonals"
Ashley Lewis
CTSE 5040
September 30, 2011
How to Sketch the Midsegment of a
Trapezoid
The midsegment of a trapezoid is best described as a line segment connecting the
midpoint of one leg, which is a side not parallel to the base nor is it the base itself, to the
midpoint of the other leg in the figure. There are two types of trapezoids, arbitrary and
isosceles, but after both trapezoids are sketched, the same steps apply to finding the
midsegment. The following is a step by step description and sketch, created using
Geometric Sketchpad, of how to find the midsegment of an arbitrary trapezoid.
First, draw a line segment
arbitrary point labeled C.
labeled AB and then draw an
Next, construct a
segment AB that is
line parallel to the line
also passing through C.
Then, connect the set of parallel lines by creating
and AD with D being a new point on the line
the line segments AC
passing through C.
Now that the steps for constructing an arbitrary trapezoid have been shown, the next
figure to create is the isosceles trapezoid.
The first step will be the same as the first step in creating an arbitrary trapezoid; draw a
line segment labeled
AB and also draw
an arbitrary point
labeled C.
The second step is the same as well. Select the line segment AB and also the point C
then select the construct a parallel line. This line will pass through C and be parallel to
line segment AB.
The third step is where the creation of an arbitrary and isosceles trapezoid begin to
differ. For this specific type of trapezoid select the points A, B, and C in that order.
Then select the arch through 3 points option under the construction menu.
Next, place a point at the intersection of the arc and the line parallel to AB and label it D.
Also, draw a line segment connecting point A and C and draw another line segment
connecting point B to
point D.
Now, that an arbitrary trapezoid and an isosceles trapezoid have been constructed the
steps to finding the midsegment are the same.
First, highlight the
select construct
will appear on the
and the other F.
segments AC and BD and
midpoints. The midpoints
lines. Label one point E
Then construct a line segment connecting point E to F. This line segment EF is known
as the midsegment of the trapezoid.
How to Sketch the Diagonals of a
Trapezoid
To sketch the diagonals of a trapezoid connect the points A and D with a line segment.
Also, connect
points B and C with a
line segment.
Information and Explanations about the
Midsegment of a Trapezoid
There are many interesting facts about a trapezoid's midsegment, or median as it is
often times called. One of the most obvious truths concerning the midsegment is that it
runs parallel to the bases of the trapezoid. Another detail that will always hold true of
the midsegment is that it is found halfway between the two bases; the reason for this is
because the midsegement is a connection of the midpoints on each leg of the trapezoid.
Another fact is the midsegment divides the original trapezoid into two smaller
quadrilaterals that are also trapezoids. In this case the midsegment acts as a base for
each of the smaller trapezoids. The measurements of the midsegment are also
important information. The
length of a trapezoid's midsegment can
be found using the formula:
In other words, the midsegment of a trapezoid can be measured by averaging the
length of the two bases. Additional information about a trapezoid can be discovered by
using the midsegement. The area of a trapezoid is equal to the height multiplied by the
length of the midsegment.
Information and Explanations about the
Diagonals of a Trapezoid
The diagonals of a trapezoid have many properties. One of the most well known is that
each diagonal cuts the trapezoid into two triangles. The isosceles trapezoid in particular
has diagonals of equal length which means they are congruent. This is true because it's
base angles have equal degree measurements. The length of a trapezoid has a
specific formula. This illustration and formula are both found on the wolfram alpha
website.
Information and Explanations about the
Segment Between the Diagonals of a
Trapezoid
In addition to the midsegment and diagonals of a trapezoid another line segment has
properties that need to be discussed, the line segment between the two diagonals. This
can be drawn by constructing a trapezoid with both the midsegment and the diagonals
in place. The portion of the
midsegment that is
between, or connecting, the
diagonals is the
segment being referred to.
The following is a
picture to help illustrate.
The length of segment GH can be found by manipulating information that is already
given. It is known that triangle AEG is similar to triangle ACD with a ratio of 1 to 2,
respectively; this is due to the fact that segment AE is halfway to segment AC because
E is the midpoint From this it can be said that segment EG is one half the length of CD.
The same can also be said for HF. Now to find the length of GH subtract the lengths of
(EG + HF) from EF.
Classroom Implications
In our CTSE 5040 class we have discussed the properties of a trapezoid. In fact, our
class participated in a discussion about the actual definition of a trapezoid. The
conclusion, that won with majority vote, was a trapezoid is a quadrilateral with at least
one set of parallel sides. By using Sketchpad this semester I have been able to discover
the properties of many figures on my own; whereas before, I would have needed
additional resources to come to a conclusion. If I were a geometry teacher lecturing on
the properties of a trapezoid I would allow my students time to use Sketchpad and see
what information they discover. I would also use Sketchpad to show students how the
properties hold true for any arbitrary triangle.